Can linear least squares be used for inverse function approximation?

[xactmetric]

Hi,

Forgive me if the subject of this post is not accurate, I'm not quite sure what the correct terminology would be for what I'm trying to figure out.

Currently I am using linear least squares via SVD to find the coefficients of a ten term polynomial, say f. This model allows me to predict some output y given some input x. Thats straightforward.

What I would like to be able to do is turn this around and find a function g where g would take an input x, and output some y' such that f( y' ) = x.

Is this familiar to anyone? I'm totally stumped.

xactmetric

 

[HallsofIvy]

Why can't you do that now? In linear least squares, you are given a list of points, [itex]{(x_1, y_1), (x_2, y_2), \cdot\cdot\cdot, (x_n, y_n)}[/itex] and you construct an equation y= ax+ b whose graph passes close (in the least squares sense) to each of those points. Given any other x, the "predicted" y value is ax+ b. But linear equations are easily solved. Given any y, the "predicted" x value is just (y- b)/a. All the work has already been done in using linear least squares to find a and b.

 

[xactmetric]

Yes but my equation is nonlinear. Let me try and explain better.

Originally I start with two lists of points. Let's put them all on the x-axis for simplicity. The first list I will call actual coordinates. The second list I will call deviated coordinates. The deviated coordinates are simply the actual coordinates plus some offset. My prediction function f just predicts the deviation of an inputted actual coordinate.
So if I input any actual coordinate x into f, f will give me a predicted deviation. So my predicted deviated point is x + f(x).

Now say I know in advanced a point x. I need to find a way to determine a point x' such that f( x' ) gives me the deviation I need such that x' + f( x' ) = x.

Does this make sense? Or am I missing something totally obvious?

xactmetric

 

[John Creighto]

I don't think you are using the right terminology. I would think for inverse least squares you might look at something like this:

http://signals.auditblogs.com/2007/07/05/multivariate-calibration/

Anyway, so basically what it sounds like is you fit a polynomial to a lower order polynomial using least squares. You have two choices. You can either find the roots of this lower order polynomial, or instead you can do a new least squares on the inverse function.

 

[xactmetric]

Thanks, but can you please elaborate a bit. I'm not following you.

xactmetric

 

[John Creighto]

Thanks, but can you please elaborate a bit. I'm not following you.

xactmetric

Can you first try to write down the exact problem you are trying to solve clear.

 

[xactmetric]

The problem is exactly as described in my second post. If you can say what's not clear, I'll try and explain a bit more.